Can someone explain the process of finding c(n) in an infinite well?

[iScience]

we're learning about some of the properties of the steady state wave functions confined in an infinite well. one of the properties was that the steady state wave functions are "complete". and we're learning how to find the coefficient c(n) that "weights" each steady state solution in finding the general solution.

http://i.imgur.com/p0Ewvx8.jpg

can someone please explain to me what exactly I'm doing in the integral in order to find c(n)?

first of all, why am i even doing anything with ψ(m)*? ie how does the 'ψ(m)' come into play? what IS it?? is it just another steady state wave function?

so then in the end is this saying that the coefficient for the nth steady state solution is equal to the steady state solution of ... the orthogonal steady state ψ(m)?..

 

[jtbell]

To understand what's happening here, it might help to pretend that there are only a small number of steady-state wave functions, say 4 instead of ∞, then choose a value of m (i.e. one particular coefficient c_m to evaluate) and write out the sum explicitly.

 

[iScience]

i understand the part where the other values go to zero, I'm more confused with the bigger picture; the coefficient c(n) is associated with the steady state psi(n). so what does any other function have to do with c(n)? i just don't understand the role of another function with c(n)

 

[WannabeNewton]

Basically the set of steady-states $\{\psi_n \}_{n \in \mathbb{Z}}$ forms an orthonormal basis for the solution space. Whenever you see "these states form a complete orthonormal set" it's the redundant and pointless physics terminology for "these states form an orthonormal basis". The orthonormal part means $\int _{-\infty}^{\infty}\psi^{*}_m \psi_n dx = \delta_{mn}$ and the complete part means any solution $f(x)$ can be written as a unique linear combination of the $\psi_n$.

We can therefore express $f(x)$ as $f(x) = \sum c_n \psi_n$. Now if we integrate both sides with $\psi^*_m$ we get $\int_{-\infty}^{\infty} \psi^*_m f(x)dx = \int_{-\infty}^{\infty} \psi^*_m\sum c_n \psi_n dx = \sum c_n \int_{-\infty}^{\infty}\psi^*_m\psi_n dx = \sum c_n \delta_{mn} = c_m$ (ignoring the subtleties about exchanging sums and integrals) so all we're doing is using the orthonormality $\int _{-\infty}^{\infty}\psi^{*}_m \psi_n dx = \delta_{mn}$ of the set $\{\psi_n \}_{n \in \mathbb{Z}}$ in order to express the coefficients $c_n$ of the basis expansion of $f(x)$ in terms of $f(x)$ and $\psi^*_n$.

This is entirely analogous to what we do in finite dimensional inner product spaces. If you have $\vec{v} = \sum c_n \vec{v}_n$ then you say $c_n = \vec{v}\cdot \vec{v}_n$. It's the same thing here except the finite dimensional inner product (or dot product) is replaced by the integral inner product above.